Theoretical Question about Buoyancy

This is a question that's been bothering me for a while. Archimedes's principle states that the weight of the water displaced is equal to the buoyant force. If a frictionless box sinks to the bottom of a lake, does it feel a buoyant force? Buoyancy exists because of a difference in pressure between the top and the bottom of the box. If the box is already at the bottom, then there can be no water underneath the box, and consequently no pressure exists underneath the box. If this is so, then does a buoyant force NOT exist? Doesn't this go against Archimedes' principle, which to the best of my knowledge does not contain any exceptions?

If you could remove all water from underneath the box, and there was no water below it, and there was no way water could slip under it (be pressed under it by pressure) and there was no water or air in the ground under it... well, there might be a case where Archimede's principle did not apply. Maybe there is something I'm missing...

But it doesen't matter, because you will never remove all the free matter from under it. There will always be water under it, or the water can be squeezed under it, or there is water in the ground etc...

I've been informed of a certain ping-pong ball experiment, where you put a ping-pong ball into an upside-down container that's been cut open at the other end. If you then pour water on top of the ping-pong ball, the ping-pong ball will stay at the bottom.